Revisiting 𝐶 and 𝐶𝑃 violation in 𝜂 → 𝜋 + 𝜋 - 𝜋 0 decay
The decay 𝜂 → 𝜋 + 𝜋 − 𝜋 0 is an ideal process in which to study flavor-conserving 𝐶 and 𝐶𝑃 violation beyond the Standard Model. We deduce the 𝐶- and 𝐶𝑃-odd quark operators that contribute to 𝜂 → 𝜋 + 𝜋 − 𝜋 0 originating from the mass-dimension-six Standard Model effective field theory. The corresponding hadron-level operators that generate a nonvanishing 𝐼 = 0 amplitude at order 𝑝 6 in the chiral effective theory are presented for the first time, to the best of our knowledge, in addition to the leading-order operators ascribed to the 𝐼 = 2 final state. By fitting the KLOE-2 and the most recent BESIII experimental data, we determine the coefficients of the lowest-order 𝐼 = 0 and 𝐼 = 2 amplitudes and estimate the potential new physics energy scale. We also perform an impact study of the future 𝜂 → 𝜋 + 𝜋 − 𝜋 0 experiments.